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| 23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical |
| Full Idea: Naïve set theory is based on the principles that any formula defines a set, and that coextensive sets are identical. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.2) | |||
| A reaction: The second principle is a standard axiom of ZFC. The first principle causes the trouble. |
| 23447 | In classical semantics singular terms refer, and quantifiers range over domains |
| Full Idea: In classical semantics the function of singular terms is to refer, and that of quantifiers, to range over appropriate domains of entities. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 7.1) |
| 23443 | The axioms of group theory are not assertions, but a definition of a structure |
| Full Idea: Considered in isolation, the axioms of group theory are not assertions but comprise an implicit definition of some abstract structure, | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.5) | |||
| A reaction: The traditional Euclidean approach is that axioms are plausible assertions with which to start. The present idea sums up the modern approach. In the modern version you can work backwards from a structure to a set of axioms. |
| 23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms |
| Full Idea: Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.1) | |||
| A reaction: This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds. |
| 23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory |
| Full Idea: If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.6) | |||
| A reaction: Note that this concerns consistency, not completeness. |
| 23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world |
| Full Idea: Philosophical structuralism holds that mathematics is the study of abstract structures, or 'patterns'. If mathematics is the study of all possible patterns, then it is inevitable that the world is described by mathematics. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 11.1) | |||
| A reaction: [He cites the physicist John Barrow (2010) for this] For me this is a major idea, because the concept of a pattern gives a link between the natural physical world and the abstract world of mathematics. No platonism is needed. |
| 23441 | Logical truth is true in all models, so mathematical objects can't be purely logical |
| Full Idea: Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 2) | |||
| A reaction: Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that). |
| 23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics |
| Full Idea: Game Formalism seeks to banish all semantics from mathematics, and Term Formalism seeks to reduce any such notions to purely syntactic ones. | |||
| From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.3) | |||
| A reaction: This approach was stimulated by the need to justify the existence of the imaginary number i. Just say it is a letter! |